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  • Petr Krysl

    A Pragmatic Introduction toFinite Element Analysisfor Structural Engineers

    With the Matlab toolbox SOFEA

    November 2005

    Pressure Cooker PressSan Diego

  • Contents

    Part I Introducing the Galerkin method

    1 Model of a Taut Wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Deriving the PDE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Boundary conditions (in space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Anything else? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2 The method of Mr. Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Residual of the balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Integral test of the residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Trial function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Manipulation of the residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Stiffness and mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Piecewise linear basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Numerical quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 Putting it together: system of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    3 Introducing the Matlab code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Statics: uniform load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Free vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Virtual work principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    Part II Thermal analysis

    4 Model of Heat Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Constitutive equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 Initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Summary of the PDE model of heat conduction . . . . . . . . . . . . . . . . . 29

  • VI Contents

    5 Galerkin method for the model of heat conduction . . . . . . . . . . . . 315.1 Weighted residual formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Reducing the model dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Test and trial functions: basis functions on triangulations . . . . . . . . . 345.4 Basis functions on the standard triangle . . . . . . . . . . . . . . . . . . . . . . . . 355.5 Discretizing the weighted residual equation . . . . . . . . . . . . . . . . . . . . . 375.6 Derivatives of the basis functions; Jacobian . . . . . . . . . . . . . . . . . . . . . 405.7 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.8 Conductivity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.9 Surface heat transfer matrix and load . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    6 Steady-state heat diffusion solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1 Steady-state diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Thick-walled tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Orthotropic insert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 The T4 NAFEMS Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

    7 Transient heat diffusion solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.1 Discretization in time for transient heat diffusion . . . . . . . . . . . . . . . . 617.2 Transient diffusion: The T3 NAFEMS Benchmark . . . . . . . . . . . . . . . 637.3 Transient cooling in a shrink-fitting application . . . . . . . . . . . . . . . . . . 65

    8 Expanding the library of element types . . . . . . . . . . . . . . . . . . . . . . . . 698.1 Quadratic triangle T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.2 Quadratic 1-D element L3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.3 Point element P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 718.4 Measuring (integrating) over domains . . . . . . . . . . . . . . . . . . . . . . . . . . 728.5 On the simplex elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748.6 Quadrilateral Q4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.7 Tetrahedron T4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

    9 Convergence and error control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779.1 First look at errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779.2 Richardson extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789.3 The T4 NAFEMS Benchmark revisited . . . . . . . . . . . . . . . . . . . . . . . . . 789.4 Shrink fitting revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    Part III Stress analysis

    10 Model of elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8310.1 Balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

    References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

    Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

  • Part I

    Introducing the Galerkin method

  • 1

    Model of a Taut Wire

    This chapter will formulate a relatively simple model of a taut string. In the nextchapter, we will seek approximate solutions to this model that are obtained withthe Galerkin method.

    1.1 Deriving the PDE model

    Figure 1.1 illustrates an idealization of a taught wire. The wire is under prestressingforce, P , assumed to be uniform along the length of the wire. The left hand end isimmovably fixed, while the right hand side end is held in a fixture which can slideperpendicularly to the axis of the wire. A transverse force FL is applied at the mov-able end. In addition, there may be some distributed force q acting along the length(but we shall ignore gravity). The transverse displacement is a function of both theaxial coordinate x and the time t, w = w(x, t) . The transverse displacement isassumed to be very small compared to the length of the wire.

    Fig. 1.1. Schematic of taut wire

    1.2 Balance equation

    Taking a section of length x of the wire (see Figure 1.2, collecting all the forces,and equating them to the inertial force (Newtons law), leads to a balance equationfor the taut wire

    P2w

    x2+ q = w , (1.1)

  • 4 1 Model of a Taut Wire

    where w = 2w

    t2 is the acceleration.

    Fig. 1.2. The forces acting on a segment of the taut wire

    1.3 Boundary conditions

    The function w that describes the transverse deflection takes two arguments, x,and t. It is defined on a rectangle shown in Figure 1.3: 0 x L, and 0 t t.It needs to be determined to satisfy the balance equation (1.1), but that wouldnot completely nail the answer down. We also know something about the solution,namely at the boundaries of the domain rectangle.

    How many pieces of information do we need to know? A reasonable answer is,Enough to make the solution unique. To find the deflection w is going to involveintegration, because the balance equation refers to space and time derivatives of w.Using the definitions

    v =w

    t

    =w

    x

    we may rewrite all the balance equation that involves the second derivatives of thefunction w as a system of first order differential equations

    t=

    v

    x

    T

    x+ q v

    t= 0

    For each derivative vx ,x , one boundary condition (integration constant) will

    be needed. Similarly, for each of the time derivatives vt , andt one boundary

    condition along the time axis will be required.

    1.4 Boundary conditions (in space)

    The conditions on w along the edges of the domain rectangle parallel to the timeaxis are known (for historical reasons) as the boundary conditions. (Perhaps alsobecause they are applied along the physical boundaries of the structure.)

    At the left-hand side end of the wire we are prescribing in general nonzerodisplacement